### Abstract

Let P be a convex polygon. A great many papers are devoted to investigation of various random values associated to the finite samples from the uniform distribution in the polygon, such as the number of the vertices of the convex hull of the sample, its circumference, probability that th convex hull has the fixed number of vertices, etc. In this note we address an inverse problem - whether and to what extent the distributions of these random values determine P. We show that Sylvester numbers, that is, the probabilities that the convex hull of m random point is a triangle for m = 4, 5, . . . determine generic polygon P unambiguously (up to affine transformations). Some general constructions, conjectures, and corollaries are presented.

Original language | English |
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Pages (from-to) | 101-116 |

Number of pages | 16 |

Journal | Advances in Applied Mathematics |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1996 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Cite this

*Advances in Applied Mathematics*,

*17*(1), 101-116. https://doi.org/10.1006/aama.1996.0005

**Counting the shape of a drum.** / Baryshnikov, Yu.

Research output: Contribution to journal › Article

*Advances in Applied Mathematics*, vol. 17, no. 1, pp. 101-116. https://doi.org/10.1006/aama.1996.0005

}

TY - JOUR

T1 - Counting the shape of a drum

AU - Baryshnikov, Yu

PY - 1996/3

Y1 - 1996/3

N2 - Let P be a convex polygon. A great many papers are devoted to investigation of various random values associated to the finite samples from the uniform distribution in the polygon, such as the number of the vertices of the convex hull of the sample, its circumference, probability that th convex hull has the fixed number of vertices, etc. In this note we address an inverse problem - whether and to what extent the distributions of these random values determine P. We show that Sylvester numbers, that is, the probabilities that the convex hull of m random point is a triangle for m = 4, 5, . . . determine generic polygon P unambiguously (up to affine transformations). Some general constructions, conjectures, and corollaries are presented.

AB - Let P be a convex polygon. A great many papers are devoted to investigation of various random values associated to the finite samples from the uniform distribution in the polygon, such as the number of the vertices of the convex hull of the sample, its circumference, probability that th convex hull has the fixed number of vertices, etc. In this note we address an inverse problem - whether and to what extent the distributions of these random values determine P. We show that Sylvester numbers, that is, the probabilities that the convex hull of m random point is a triangle for m = 4, 5, . . . determine generic polygon P unambiguously (up to affine transformations). Some general constructions, conjectures, and corollaries are presented.

UR - http://www.scopus.com/inward/record.url?scp=0030098888&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030098888&partnerID=8YFLogxK

U2 - 10.1006/aama.1996.0005

DO - 10.1006/aama.1996.0005

M3 - Article

AN - SCOPUS:0030098888

VL - 17

SP - 101

EP - 116

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

IS - 1

ER -